Journal Article

Frobenius distributions of elliptic curves over finite prime fields

Ernst-Ulrich Gekeler

in International Mathematics Research Notices

Volume 2003, issue 37, pages 1999-2018
Published in print January 2003 | ISSN: 1073-7928
Published online January 2003 | e-ISSN: 1687-0247 | DOI: https://dx.doi.org/10.1155/S1073792803211272
Frobenius distributions of elliptic curves over finite prime fields

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We study the number H(t,p) of isomorphism classes of elliptic curves with Frobenius trace t ∈ ℤ over the finite field Fp. It may be predicted through a probabilistic model based on the frequencies of matrices in GL(2,ℤ) with characteristic polynomial X2-tX+p, for primes ℓ ≠ p. We describe the asymptotics of the summatory function Ht(x) = ∑ H(t,p) (where p runs through the primes less than x ∈ ℝ). It turns out that the likelihood of a given t ∈ ℤ as a Frobenius trace is proportional to w(t)=∏ ((ℓ2−ℓ)/(ℓ2−ℓ−1))(ℓ|t,ℓ prime).

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Subjects: Mathematics

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